Question: For how many real values of $c$ do we have $\left|\frac12-ci\right| = \frac34$?
We have $\left|\frac12-ci\right| = \sqrt{{\frac12}^2 + (-c)^2} = \sqrt{c^2 + \frac14}$, so $\left|\frac12-ci\right| = \frac34$ gives us $\sqrt{c^2 + \frac14} = \frac34$.  Squaring both sides gives $c^2 + \frac14 = \frac9{16}$, so $c^2=\frac5{16}$.  Taking the square root of both sides gives $c = \frac{\sqrt5}4$ and $c=-\frac{\sqrt5}4$ as solutions, so there are $\boxed{2}$ real values of $c$ that satisfy the equation.

We also could have solved this equation by noting that $\left|\frac12-ci\right| = \frac34$ means that the complex number $\frac12-ci$ is $\frac34$ units from the origin in the complex plane.  Therefore, it is on the circle centered at the origin with radius $\frac34$.  The complex number $\frac12-ci$ is also on the vertical line that intersects the real axis at $\frac12$, which is inside the aforementioned circle.  Since this line goes inside the circle, it must intersect the circle at $\boxed{2}$ points, which correspond to the values of $c$ that satisfy the original equation.